This new type of dice guarantees no tie when deciding who goes first

During a dinner conversation at a gaming convention in 2012, Eric Harshbarger was asked by a board game designer if he could come up with dice that would determine who goes first — without the possibility of a tie.

The idea was simple: settle the first turn quickly and get on with the game.

“The goal [is] to save everyone a few precious seconds so they can get playing their board game as quickly as possible,” Harshbarger told As It Happens host Nil Köksal.

Harshbarger, a mathematician and senior lecturer at Auburn University in Auburn, Alabama, says the problem was right up his alley. 

“It was a question that did not have an obvious answer and that’s something that a mathematician will often jump at.”

Theoretically, Harshbarger had thought up a set of dice where every player could roll one die and have an equal chance of winning, while also guaranteeing that no two players would tie.

But finding dice that actually worked in the real world proved much harder.

Now, nearly 15 years later, Harshbarger and his colleagues have finally invented a set of dice — the Go First Dice for five players — that can allow them to decide fairly who moves first in a board game, with a winner guaranteed from a single roll.

A problem that stuck around

Over the years, Harshbarger became a steward for the problem, operating a website to track discoveries as about 20 computer scientists and mathematicians from around the world drifted in and out of the project.

It didn’t take long for Harshbarger and his team to work out a set of four 12-sided dice. The dice allowed two, three or four players to each throw one of the dodecahedrons and get a statistically fair result, with a winner every time.

“With two people, we had that answered in the blink of an eye, with three people we had answered in a day or two, four people we had the dice within two to three weeks, maybe a month,” said Harshbarger. 

But five players turned out to be a much tougher puzzle.

Harshbarger says he and his colleagues always knew the dice were mathematically possible.

The mystery was whether that mathematical solution could be translated into the physical geometry of a die — something that could actually be manufactured and rolled.

“I knew there was a solution with something crazy like 1,440 sides for each die,” he said. “That’s not makeable.”

So the researchers began developing different theories and trying to reduce the number of sides, gradually “whittling things down.”

During that process, a researcher in Australia joined the group and came up with a more practical solution: a set of five 120-sided dice.

The discovery generated publicity, and within about six months, another researcher would join the effort — this time, from Canada. 

A breakthrough from Canada

Paul Meyer, a Canadian software engineer, reached out to Harshbarger and asked for some of his old results. 

Three months later, Meyer emailed Harshbarger with exciting news: he had found a solution using five 60-sided dice. 

Throughout the project, many people emailed Harshbarger claiming they had solved the problem, so his first step was always to verify the proposed solution. 

This time, it actually worked.

Harshbarger emailed Meyer back: “Do you understand that we’ve been looking for this for a long time?”

Meyer had approached the problem as a computational challenge, using mathematical patterns from previous solutions and a brute-force computer search.  

The search was enormous — there were more possible designs than atoms in the universe — but he narrowed it down enough to make the discovery. 

The golf-ball-sized dice are now available to buy, but some researchers have questioned whether they can truly live up to the mathematical theory in practice.

Harshbarger, however, says the mathematics are sound. The dice are designed to be perfectly fair, and any imperfections that could affect the outcome would come from the manufacturing process, not the underlying mathematics.

The next unanswered questions

Meyer is now part of the research group, and their work is ongoing.

Harshbarger says they are still exploring whether fewer sides could work for a five-player set, with 30 sides known to be the minimum, and whether the concept could be expanded to six players.

Their current mathematical and computational tools aren’t enough to solve those problems, he says, but they’ve learned to expect the unexpected. 

“Definitely the door’s open that someone else could walk in with a brand new theory and say, ‘well this is what I did,’” said Harshbarger. “That’s pretty much what happened with Paul. He came out of the blue.”

For Harshbarger, that unpredictability is part of what made the problem so compelling in the first place.

“Mathematically, it seemed very beautiful and very intriguing even though we knew from day one, it was a rather silly question to ask.”

“But it was a question that did not have an obvious answer and that’s something that a mathematician will often jump at and stay with doggedly for years and years.”